arX

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v1 [

cond

-mat

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22

Aug

201

4Preprint option

The Effect of Dissipation on the Torque and Force Experienced

by Nanoparticles in an AC Field

F. Claro∗

Pontificia Universidad Catolica de Chile, Casilla 306, Santiago, Chile

R. Fuchs†

Ames Laboratory and Iowa State University, Ames, Iowa 50011, USA

P. Robles‡

Escuela de Ingenierıa Electrica, Pontificia Universidad

Catolica de Valparaıso, Casilla 4059, Valparaıso, Chile

R. Rojas§

Universidad Tecnica Federico Santa Marıa, Casilla 110-V, Valparaıso, Chile

Abstract

We discuss the force and torque acting on spherical particles in an ensemble in the presence of

a uniform AC electric field. We show that for a torque causing particle rotation to appear the

particle must be absorptive. Our proof includes all electromagnetic excitations, which in the case

of two or more particles gives rise to one or more resonances in the spectrum of force and torque

depending on interparticle distance. Several peaks are found in the force and torque between two

spheres at small interparticle distances, which coalesce to just one as the separation grows beyond

three particle radii. We also show that in the presence of dissipation the force on each particle

is non conservative and may not be derived from the classical interaction potential energy as has

been done in the past.

∗Electronic address: fclaro@uc.cl†Electronic address: fuchs@iastate.edu‡Electronic address: probles@ucv.cl§Electronic address: roberto.rojas@usm.cl

1

I. INTRODUCTION

The electromagnetic excitations and ensuing dynamics of nanoparticles, molecules and

atoms in the presence of an electric field has been widely studied both theoretically and

experimentally [1–5]. The particles may initially be unpolarized, but due to the external field

and their mutual interaction they may acquire induced dipole and higher electric moments.

As a consequence electric forces and torques are produced, resulting in particle motion

and the formation of equilibrium configurations. An important case is optical trapping

and binding, which, if many particles are involved may lead to self-assembly of ordered

structures [6]. Structures are also formed in electro-rheological fluids, where a static or slowly

varying field induces the formation of linear arrays and columns in a medium containing

polarizable spheres in suspension [7]. Examples where an understanding of forces and torques

is also crucial are the dielectrophoresis and electrorotation effects, related to motion in a

non uniform field [8] and a rotating AC field [9], respectively. Other applications include the

control of agglomeration, and the separation of proteins or living cells in suspension [10].

Nanorotors driven by a light force have also been studied [11, 12].

Several methods have been used to obtain forces [13–16] and torques [17–20] in the past,

some involving the use of an interaction potential energy whose gradient is taken to obtain

the force [13, 21–23]. In this work we prove that if the particles are absorptive the system

is non-conservative and the net force experienced by each member of the ensemble may not

be derived from an interaction potential. In fact, we show explicitly that structure in the

interaction energy arising from absorption resonances in a pair of gold nanospheres exhibits

energy minima leading to unphysical equilibrium configurations that are not present if the

force is calculated directly from Coulomb’s law [23, 24]. Away from such resonances when

absorption is negligible either method may be used leading to similar results.

In order to obtain explicit expressions for the force and torque we asume the particles to

be spherical, thus allowing a multipolar analysis and a comparison with results obtained in

the dipole approximation. For an AC external excitation we find the dipole approximation to

give accurate results if the center to center separation between neighbors is not smaller than

three particle radii, while at closer interparticle distance the inclusion of all multipoles gives

rise to several resonances in the force and torque strength, shifted to lower frequencies owing

to particle-particle couplings. This is in accordance with previous results on the electric

2

excitation of dielectric particles arrays showing a similar distance dependent behavior [25–

28]. Location of such resonances in the frequency spectrum may be useful in applications

when the force or torque strength becomes important. Within the same model we find that

the appearance of a torque causing particle rotation requires that the particle be dissipative.

The paper is organized as follows. In Sec. II we present compact expressions for the time-

averaged force and torque acting over a particle in an arbitrary array of nanoparticles in a

uniform AC electric field. The very structure of the resulting expressions reveals the need

for dissipation in order for a torque to arise. The cases of linear and circular polarization are

discussed. In order to assess the relevance of higher multipoles in both forces and torque,

in Sec. III we apply our model to two gold nanospheres in an electric field parallel or

perpendicular to the interparticle axis. In Sec. IV we prove that the presence of dissipation

makes the system non conservative, and in Sec. V we present our conclusions. Finally, two

appendices are added to provide details of the calculations.

II. FORCES AND TORQUES ON INTERACTING PARTICLES IN AN AC FIELD

We consider a system of nanoparticles embedded in a non absorptive dielectric medium,

excited by an external AC electric field of angular frequency ω. The particles are uncharged

and their material response to a local electric field may in general be characterized by

a complex response function ǫ(ω). The external field induces a dipole moment on each

particle, which in turn excites multipoles on every other member of the ensemble owing to

the non uniformity of the electric field it produces at each particle site. For simplicity we

shall assume in what follows that the particles are of spherical shape.

As known, for a dilute system with average center-to-center separation of the order of

three times the particle radii or more, the accuracy of the dipole approximation is accept-

able and the effect of higher multipoles may be neglected [25]. In such case and if only

two particles are present, the electric force between them may be simply obtained by direct

application of the discrete form of Coulombs law, as described in reference [24]. When sepa-

rations less than three particle radii become involved however, the effect of higher multipoles

must be included [26, 27]. The general form of Coulomb’s law to be used is then,

⟨

~Fi

⟩

=1

2Re

∫

ρ∗i (~r)~E (~r) d3~r , (1)

3

where⟨

~Fi

⟩

is the time-averaged force on particle i, ρ∗i (~r) is its charge density and ~E (~r) is

the local electric field due to the external sources and other particles in the ensemble. A

rather lengthy calculation then yields the force cartesian components (see Appendix A),

〈Fix〉 = Re∑

l

CliReTli , (2)

〈Fiy〉 = Re∑

l

CliImTli , (3)

〈Fiz〉 = Re∑

l

Cli

l∑

m=−l

√

(l −m)(l +m)qlmiq∗l−1,m,i , (4)

where the pole order index l here and in what follows covers the range of integers 1,∞. In

the above expressions the coefficient

Cli =2π

√

(2l + 1)(2l − 1)αli

(5)

weights the strength with which the multipole of order l contributes, with αli the corre-

sponding particle polarizability, a complex quantity if absorption is present. Also

qlmi =∫

ρi (~r) rlY ∗

lm (θ, φ) d3~r (6)

is the induced multipole of indices l, m on particle i. and

Tli =∑

m

√

(l −m)(l −m− 1)qlmiq∗l−1,m+1,i . (7)

Ylm is the usual complex spherical harmonic function. Methods to obtain the multipoles

qlmi for arbitrary configurations are described in Refs. [26] and [27]. Notice that since the

force involves products of multipoles of different order, if there is a single spherical particle

and the external field is uniform only the dipole moment is excited and the force is zero.

Spinning of coupled particles in an external field has been observed in the past [20, 29].

In order to capture this effect we consider next the time-averaged torque on sphere i due to

the local field, as given by

〈~τi〉 =1

2Re

∫

ρ∗i (~r)~r× ~E (~r) d3~r . (8)

where the origin is taken at the particle center. Work similar to that done above for the

forces (see Appendix B) leads to the time-averaged torque cartesian components

4

〈τix〉 = Im∞∑

l=1

DliReSli , (9)

〈τiy〉 = Im∞∑

l=1

DliImSli , (10)

〈τiz〉 = Im∞∑

l=1

Dli

l∑

m=−l

m |qlmi|2 , (11)

where the coefficients

Dli =2π

(2l + 1)αli(12)

are complex if αli is, and

Sli =l−1∑

m=−l

√

(l −m)(l +m+ 1)qlmiq∗l,m+1,i . (13)

It is clear from Eqs. (9) to (13) that if the system has no dissipation, i.e. if αli is real, the

torque is zero. We conclude that in general a torque arises in such systems from dissipative

electromagnetic interactions.

Even if there is dissipation however, the torque may be suppressed by special symmetries.

Such is the case for a linear array subject to a uniform electric field parallel to the line joining

their centers. By choosing the z-axis to be aligned with this line, only modes with m = 0

are excited leading to zero torque, as may be easily verified from the structure of the above

equations. A similar situation occurs if the applied electric field lies on the xy plane since in

this case only modes with m = ±1 are excited symmetrically and the torque is again zero.

Nevertheless, it is worth noting that if the linear array is under a uniform electric field with

components along the z-axis and the xy plane, modes with m = ±1 and m = 0 become

excited. So, according to Eqs. (9) to (11) a torque is produced provided that electromagnetic

dissipation is not negligible. A similar situation has been analyzed in Ref. [20] in the dipolar

approach.

A torque does arise in such arrays also if they are subject to a rotating electric field on the

xy plane. The field may be written as ~E = E0(±x−iy)eiωt and the corresponding coefficients

of expansion of the potential are either V1,+1 =√

2π/3E0(1− i) or V1,−1 =√

2π/3E0(−1− i)

depending of the sense of rotation of the electric field vector given by the sign of the x

component [30]. Correspondingly the excited modes are either m = 1 or m = −1 and from

5

Eq. (11) it follows that a torque may appear. In fact, from Eqs. (A35) and (11) it can be

shown that for this case the time-average of the z−component of the torque is given by

〈τiz〉 =2πm

a2l+1

∑

l

l Im ǫ

[l (Re ǫ− 1)]2 + [l Im ǫ]2|qlmi|2 . (14)

For the special case of a single sphere in a rotating external field the torque is finite, in

agreement with Refs.[31] and [32]. The physical origin of such a torque is conservation of

angular momentum. The rotating field carries angular momentum, which is transferred

to the particles when absorption takes place causing them to experience a spinning torque.

Also, as noted in Ref. [20] when a linearly polarized field is not aligned with a symmetry axis

of a linear array such as a pair, the local field at each particle site has a rotating component,

and the same argument applies.

III. SPECIAL CASE: TWO PARTICLES

We shall apply our general results to the simplest case, that of two identical spheres of

radii a subject to a uniform oscillating electric field, both parallel and perpendicular to a

line joining the spheres centers, that we choose to be the z axis. These conditions will be

referred to as parallel and perpendicular excitation, respectively. In computing the force we

found convenient to use Eq. (A33) in Appendix A with the replacement Vlmi = blmi, since

the uniform external field produces no direct force. Using relation (A7) then leads to,

〈Fiz〉 = −1

2Re

∑

lm

∑

l′m′

∑

j 6=i

(−1)l′

Al′m′jlmi

√

√

√

√

(2l + 1)

(2l − 1)(l −m)(l +m)ql′m′jq

∗l−1,m,i , (15)

where the coefficient Al′m′jlmi that couples multipoles in different particles is given by Eq. (A8)

in Appendix A.

A. Parallel excitation

In this geometry ~E = E0eiωtz and modes with m = 0 become excited only, yielding a

force along the z-axis. From Eqs. (2), (3) and (7) it is seen that the time-averaged value

of the components x and y of the force is zero, as expected from symmetry considerations.

6

Using Eq. (15) we find after some algebra the force component on sphere 1 centered at the

origin

〈F1z〉 = −2πRe∑

ll′(−1)l

′ (l + l′ + 1)!

l!l′!√

(2l + 1) (2l′ + 1)Rl+l′+2q∗l,0,1ql′,0,2 , (16)

where the multipole moments may be obtained using the formalism of Ref.[27]. Here R is

the center to center distance between the two spheres. The dipole approximation applies

keeping the first term in this series,(l = l′ = 1), and the result agrees with that in Ref.[24]

as it should.

B. Perpendicular excitation

In this case the external field is in the xy plane, and the external potential in Eq. (A10)

of Appendix A is expressed as V ext = V1,1,irY1,1 (θ, φ) + V1,−1,irY1,−1 (θ, φ) with V1,±1 =√

2π/3 (±Ex − iEy). The coupling coefficients in Eq. (A8) are null unless m = m′ = ±1.

From Eq. (15) we get this time,

〈F1z〉 = 2πRe∑

ll′(−1)l

′ (l + l′ + 1)!

l!l′!Rl+l′+2

√

ll′

(2l + 1) (2l′ + 1) (l + 1) (l′ + 1)

[

q∗l,1,1ql′,1,2 + q∗l,−1,1ql′,−1,2

]

.

(17)

Keeping just the l = l′ = 1 term in the series the dipole approximation is obtained, which

agrees with the corresponding expression in Ref.[24].

C. Numerical Results

We next show some numerical results for our test case of two particles. We use a Drude

dielectric function with parameters ǫb = 9.9, hωp = 8.2 eV, Γ = 0.053 eV, appropriate

for gold nanospheres [33]. Figure 1 shows the average force for parallel (solid line) as well

as perpendicular (dashed line) excitation. One particle is at the origin, while the other is

at z = R. The separation is R = 2.005a and we have included multipoles up to order

L = 40 in the computation, following the convergence criterion given in Ref. [26]. The

force acting on the particle at the origin is attractive (positive) in the parallel configuration

and repulsive (negative) in the perpendicular geometry, as expected. Three multipolar

7

resonances are clearly resolved at this separation, with force peaks greatly enhanced, about

three orders of magnitude above the background value. As the separation between the

particles is increased the resonances move to higher frequencies, decrease in size and fewer

of them become resolved [27]. At a center to center separation of about three particle radii

and larger, only one resonance is seen. This dipolar peak, at separation R = 3a and parallel

excitation, has been included in the figure for comparison with an amplification factor of

one thousand (dash-dotted curve).

In Figure 2 we show the z component of the average torque acting on each nanoparticle as

given by Eq. (14). Separations are R = 2.005a (solid curve) and R = 3a (dashed curve). The

pair is subject to an electric field whose direction rotates in the plane xy. As for the force,

several resonances are resolved at small separation, while beyond about separation R = 3a

only one peak is observed. It can be seen that as the spheres become closer other resonances

occur at frequencies below the single sphere dipole resonance value ω = ωp/√ǫb + 2. These

additional resonance frequencies correspond to resonant modes associated with the multipole

moments qlmi.

IV. LIMITS IN THE USE OF AN INTERACTION ENERGY TO OBTAIN THE

FORCE

The existence of dissipation makes a system non conservative. To see this, recall that

for ideal electromagnetic arrays where dissipation is absent, the force acting on a particle

may be obtained as the gradient with respect to the particle coordinates of the configuration

energy W . If the particle makes a virtual displacement δξ the corresponding electric force

it is subject to is Fe = ∂We/∂ξ, an expression obtained by the energy balance equation,

δWsource = Feδξ + δWe , (18)

where δWsource is the energy supplied by the sources to maintain the potentials of the

electrodes fixed, and δWe is the variation in the energy stored in the field. It can be shown

that for this case δWsource = 2δWe so that the expression Fe = ∂We/∂ξ is obtained [30, 34].

Nevertheless, for real systems dissipation effects must be taken into account and that is

done adding a term δWloss in the right side of Eq. (18). This term depends on the path

followed during the virtual displacement since the polarization in the particle does and the

8

energy loss is determined by its imaginary part. If the particle is brought from point A to

point B, to the mechanical work done one must add the energy loss term∫ τ0 P absdt, where

P abs is the time averaged power absorbed by the system and τ the time taken during the

displacement. Both the integrand and the upper limit of this integral depend on the path

making the mechanical system non conservative.

Based on the above argument we state that in a dissipative system it is incorrect to

obtain the force as the gradient of a potential. To illustrate the difference between a direct

application of Coulomb’s law and the use of a potential we consider two polarizable spheres

of radius a, a distance R apart in an electric field of frequency ω and amplitude E0 which for

simplicity we choose to be parallel to the line joining the centers. In the dipole approximation

the interaction energy is of the form [23]

Wint(R) = U0 −1

2Re[β1(R)− β)]a3E2

0 , (19)

where U0 is the free-field interaction energy, β = (ǫ− 1)/(ǫ+ 2) with ǫ being the frequency

dependent dielectric function of the spheres, β1(R) = β/(1 − β/4σ3), and σ = R/2a. Dif-

ferentiating the second term in Eq. (19) to get the force induced by the external field we

obtain

Fw(R) = −a2E20

48σ4Re

1

(n− u)2, (20)

where n = (1 − 1/4σ3)/3 and the complex spectral variable u = 1/(ǫ − 1) has been used.

By contrast, if the direct Coulomb’s method is used one gets [24]

Fc(R) = −a2E20

48σ4

1

|n− u|2 . (21)

The two forms (20) and (21) agree only when the dielectric function is real, and dissipation

is absent. In Figure 3 we compare the force obtained using these two expressions for a pair

of gold nanospheres with a dielectric function as described en Sec. III. As can be observed

while the direct Coulomb’s method gives an attractive force at all frequencies, the model

based on the gradient of the interaction energy presents two peaks and an unphysical change

of sign in the force.

9

V. CONCLUSIONS

We have shown that in an ensemble of polarizable spheres in an oscillating electric field,

the presence of a rotation torque requires the particle material to be dissipative. We also

show that energy loss due to dissipation makes the system non conservative so that it

is improper to use an interaction energy to derive the force, an approach that has been

employed erroneously in the past [21]. Our results are an extension of previous work done

for the case of an isolated pair using the dipolar model [24]. When interparticle distances are

shorter than three particle radii it is known that the dipole approximation is not adequate,

and higher multipoles must be considered [25, 27]. Electromagnetic resonances associated

with such multipoles are known to appear, that should have a mirror spectrum in the forces

and torques as well. We have explicitely shown this to be the case in the simple case of a

pair.

Acknowledgments

During the elaboration of this paper one of the contributing authors, Professor Ronald

Fuchs of Ames Laboratory and Iowa State University, has passed away. This work is dedi-

cated to him. One of us (PR) thanks to Escuela de Ingenierıa Electrica, Pontificia Univer-

sidad Catolica de Valparaıso for its support.

10

Appendix A: Time-averaged force

We consider en ensemble of N spheres in the presence of an external electric field. Choos-

ing a coordinate system with origin at the center of particle i, the electric potential at a

point in the medium due to the polarizd spheres is given by [30]

V (~r) =∞∑

l=1

+l∑

m=−l

4π

2l + 1qlmi

Ylm (θ, φ)

rl+1+

∞∑

l=1

+l∑

m=−l

N∑

j=1

4π

2l + 1qlmj

Ylm

(

θj , φj

)

Rjl+1 , (A1)

where the multipole moment of order l, m in particle j has been defined in Eq. (6). The

center of sphere j is at ~Rj and ~r− ~Rj =(

Rj , θj, φj

)

is the position vector of the observation

point with respect to the center of sphere j. To uncouple vectors ~r and ~Rj we use the

identities [35],

Ylm(θj , φj)

Rl+1j

= (−1)l+m

[

2l + 1

4π(l +m)!(l −m)!

]1/2 [∂

∂x+ i

∂

∂y

]m∂l−m

∂zl−m

1∣

∣

∣~r − ~Rj

∣

∣

∣

, (A2)

1∣

∣

∣~r − ~Rj

∣

∣

∣

=∞∑

l=0

rl<rl+1>

4π

2l + 1

+l∑

m=−l

(−1)mYlm(θ, φ)Yl,−m(θj , φj) , (A3)

∂n

∂znYlm(θ, φ)r

l =

[

2l + 1

(2l − 2n+ 1)

(l +m)!

(l +m− n)!

(l −m)!

(l −m− n)!

]

Yl−n,m(θ, φ)rl−n , (A4)

[

∂

∂x+ i

∂

∂y

]p

Ylm(θ, φ)rl =

[

2l + 1

(2l − 2p+ 1)

(l −m)!

(l −m− 2p)!

]

Yl−p,m+p(θ, φ)rl−p . (A5)

In Eq. (A3) r<(r>) is the lower (higher) value between r = |~r| and Rj =∣

∣

∣

~Rj

∣

∣

∣; Eq. (A4) is

valid for l ≥ n and |m| ≤ l − n while Eq. (A5) is valid for l ≥ p and −l ≤ m ≤ l − 2p.

From Eqs. (A2) to (A5) and adding the potential V ext due to the external field, Eq. (A1)

becomes

V (~r) =∑

l,m

4π

2l + 1qlmi

Ylm (θ, φ)

rl+1+

∑

l,m

blmiYlm(θ, φ)rl + V ext , (A6)

where

blmi =∑

l′m′

∑

j 6=i

Al′m′jlmi ql′m′j . (A7)

11

Here Al′m′jlmi is the coupling coefficient between qlmi and ql′m′j (with i 6= j) [27]

Al′m′jlmi = (−1)m

′ Y ∗l+l′,m−m′ (θij , φij)

|Rij |l+l′+1

×[

(4π)3 (l + l′ +m−m′)! (l + l′ −m+m′)!

(2l + 1) (2l′ + 1) (2l + 2l′ + 1) (l +m)! (l −m)! (l′ +m′)! (l′ −m′)!

]1/2

,(A8)

and ~Ri − ~Rj = (Rij, θij , φij). Equations (A6) to (A8) are general and valid for any array of

spherical particles and arbitrary direction of the applied electric field. All expressions here

and below are given in Gaussian units.

In order to obtain the average force we use Eq. (1) making the replacement E (~r) =

−∇Vi(~r) for the local electric field due to the polarized system. Here

Vi (~r) =∑

lm

blmirlYlm (θ, φ) + V ext (~r) . (A9)

If we expand the external potential as

V ext (~r) =∑

lm

V extlmi r

lYlm (θ, φ) , (A10)

the above equation may be written in the form

Vi (~r) =∑

lm

VlmirlYlm (θ, φ) , (A11)

where Vlmi = V extlmi + blmi.

In order to obtain explicit expressions for the components of the force we first write the

spherical harmonics in the above equation in terms of Legendre functions using the relation

Ylm (θ, φ) =

√

√

√

√

(2l + 1)

4π

(l −m)!

(l +m)!Pml (cos θ)eimφ . (A12)

Then Eq. (A11) may be recast as

Vi (~r) =∑

lm

DlmirlPm

l (cos θ) eimφ , (A13)

where

Dlmi = Vlmi

√

√

√

√

(2l + 1)

4π

(l −m)!

(l +m)!. (A14)

12

Therefore the spherical components of the electric field are

Er = −∂Vi(~r)

∂r= −

∑

lm

Dlmilrl−1Pm

l (ξ)eimφ, (A15)

Eθ = −1

r

∂Vi(~r)

∂θ= −

∑

lm

Dlmirl−1 ∂

∂θPml (ξ)eimφ =

∑

lm

Dlmirl−1

√

1− ξ2∂

∂ξPml (ξ)eimφ(A16)

Eφ = − 1

r sin θ

∂Vi(~r)

∂φ= −

∑

lm

Dlmirl−1 1√

1− ξ2Pml (ξ)imeimφ. (A17)

In Eqs. (A15) to (A17) we have defined ξ = cosθ. The corresponding Cartesian components

of the electric field are given by

Ex = Er sin θ cosφ+ Eθ cos θ sin φ− Eφ sinφ , (A18)

Ey = Er sin θ sin φ+ Eθ cos θ cos φ+ Eφ cos φ , (A19)

Ez = Er cos θ −Eθ sin θ . (A20)

It is useful to calculate linear combinations of Ex and Ey defined as

E+ = Ex + iEy , (A21)

E− = Ex − iEy . (A22)

Introducing relations (A15) to (A19) into Eq. (A21) one obtains

E+ = −∑

lm

Dlmirl−1

[

l√

1− ξ2Pml (ξ)− ξ

√

1− ξ2∂

∂ξPml (ξ)− m√

1− ξ2Pml (ξ)

]

ei(m+1)φ .

(A23)

The relations

(1− ξ2)∂Pm

l (ξ)

∂ξ= (l +m)Pm

l−1(ξ)− lξPml (ξ) , (A24)

(l −m)Pml (ξ)− ξ(l +m)Pm

l−1(ξ) =√

1− ξ2Pm+1l−1 (ξ) , (A25)

lead then to

E+ = −∑

lm

Dlmirl−1Pm+1

l−1 (ξ)ei(m+1)φ . (A26)

Using Eq. (A14) and (A12) one obtains

E+ = −∑

lm

Vlmirl−1

√

2l + 1

2l − 1(l −m)(l −m− 1)Yl−1,m+1(θ, φ) . (A27)

Similarly, from Eqs. (A22) and (A15) to (A19) follows

E− = −∑

lm

Dlmirl−1

[

l√

1− ξ2Pml (ξ)− ξ

√

1− ξ2∂

∂ξPml (ξ) +

m√1− ξ2

Pml (ξ)

]

ei(m+1)φ .

(A28)

13

The recurrence relation ξPml−1(ξ)−Pm

l (ξ) = (l+m− 1)√1− ξ2Pm−1

l−1 (ξ) and Eq. (A24) can

be used to find that

E− =∑

lm

Vlmirl−1

√

2l + 1

2l − 1(l −m)(l +m− 1)Yl−1,m−1(θ, φ) . (A29)

To obtain Ez we use the relation (1−ξ2)∂Pm

l(ξ)

∂ξ= (l+m)Pm

l−1(ξ)− lξPml (ξ) , and Eqs. (A15),

(A16) and (A20) to give

Ez = −∑

lm

Vlmirl−1

√

2l + 1

2l − 1(l −m)(l +m)Yl−1,m(θ, φ) . (A30)

The Cartesian components of the time-averaged force acting upon sphere i are then given

by

〈Fix〉 =1

2Re

∫

ρ∗i (~r)Exd3~r =

1

2Re

∫

ρ∗i (~r)1

2(E+ + E−)d

3~r

= −1

4Re

∫

ρ∗i (~r)∑

lm

Vlmirl−1

√

2l + 1

2l − 1

×[

√

(l −m)(l −m− 1)Yl−1,m+1(θ, φ)−√

(l +m)(l +m− 1)Yl−1,m−1(θ, φ)]

d3~r

= −1

4Re

∑

lm

Vlmi

√

2l + 1

2l − 1

×[

√

(l −m)(l −m− 1)q∗l−1,m+1,i −√

(l +m)(l +m− 1)q∗l−1,m−1,i

]

, (A31)

〈Fiy〉 =1

2Re

∫

ρ∗i (~r)Eyd3~r =

1

2Re

∫

ρ∗i (~r)i

2(−E+ + E−)d

3~r

=1

2Re

∫

ρ∗i (~r)i

2

∑

lm

Vlmirl−1

√

2l + 1

2l − 1

×[

√

(l −m)(l −m− 1)Yl−1,m+1(θ, φ) +√

(l +m)(l +m− 1)Yl−1,m−1(θ, φ)]

d3~r

=1

4Re i

∑

lm

Vlmi

√

2l + 1

2l − 1

×[

√

(l −m)(l −m− 1)q∗l−1,m+1,i +√

(l +m)(l +m− 1)q∗l−1,m−1,i

]

, (A32)

〈Fiz〉 =1

2Re

∫

ρ∗i (~r)Ezd3~r

14

= −1

2Re

∫

ρ∗i (~r)∑

lm

Vlmirl−1

√

2l + 1

2l − 1(l −m)(l +m)Yl−1,m(θ, φ)d

3~r

= −1

2Re

∑

lm

Vlmi

√

2l + 1

2l − 1(l −m)(l +m)q∗l−1,m,i , (A33)

The coefficients Vlmi and qlmi are related by [26]

qlmi = −2l + 1

4παliVlmi , (A34)

where αli is the multipole polarizability of the sphere i given by [27]

αli =l(ǫ− 1)

l(ǫ+ 1) + 1a2l+1i . (A35)

We next use relation (A34) in Eqs. (A31), (A32) and (A33) to get the force components as

a sum, bilinear in the induced multipole moments. Using the property q∗l,−m = (−1)m qlm

that arises from definition (6) and the properties of spherical harmonics, one then gets

〈Fix〉 = Re∑

l

CliReTli , (A36)

〈Fiy〉 = Re∑

CliImTli , (A37)

〈Fiz〉 = Re∑

l

Cli

∑

m

√

(l −m)(l +m)qlmiq∗l−1,m,i , (A38)

where

Cli =2π

√

(2l + 1)(2l − 1)αli

(A39)

is in general a complex quantity involving the polarizability αli, and

Tli =∑

m

√

(l −m)(l −m− 1)qlmiq∗l−1,m+1,i . (A40)

The force components are thus given in compact form, convenient for numerical computation.

15

Appendix B: Time-averaged torque

In this Appendix we derive general expressions for the time-averaged components of the

torque acting upon particle i in a set of N polarizable spherical nanoparticles of radii a in

the presence of a uniform AC electric field. This is given by

〈~τi〉 =1

2Re

∫

ρ∗i (~r)~r× ~E (~r) d3~r . (B1)

The time averaged torque over particle i in the ensemble is given by Eq. (B1). The

corresponding Cartesian components are

〈τix〉 =1

2Re

∫

ρ∗i (~r)(yEz − zEy)d3~r , (B2)

〈τiy〉 =1

2Re

∫

ρ∗i (~r)(zEx − zEz)d3~r , (B3)

〈τiz〉 =1

2Re

∫

ρ∗i (~r)(xEy − yEx)d3~r . (B4)

Using the field and distance variables defined as E± = Ex ± iEy and r± = x± iy , the x and

y components of the torque can be expressed as

〈τix〉 =1

4Re

∫

ρ∗i (~r)i(W+ −W−)d3~r , (B5)

〈τiy〉 =1

4Re

∫

ρ∗i (~r)(W+ +W−)d3~r , (B6)

where W+ = zE+ − r+Ez and W− = zE− − r−Ez. From Eqs. (A26) and (A30) for E+and

Ez respectively, and introducing relation (A12) we have

zE+ = −rξ∑

lm

Dlmirl−1Pm+1

l−1 (ξ)ei(m+1)φ (B7)

r+Ez = −r√

1− ξ2eiφ∑

lm

Dlmirl−1Pm

l−1(ξ)ei(m)φ . (B8)

Eqs. (B7), (B8) and the identity −ξPm+1l−1 (ξ) + (l +m)

√1− ξ2Pm

l−1(ξ) = −Pm+1l (ξ) lead to

W+ =∑

lm

Dlmirl[

−ξPm+1l−1 (ξ) + (l +m)

√

1− ξ2Pml−1(ξ)

]

ei(m+1)φ

16

=∑

lm

Vlmi

√

√

√

√

(2l + 1)

4π

(l −m)!

(l +m)!rl

[

−Pm+1l (ξ)

]

ei(m+1)φ

= −∑

lm

Vlmirl√

(l −m)(l +m+ 1)Yl,m+1(θ, φ) . (B9)

Using Eqs. (A29) and (A30) for E−and Ez respectively, and introducing relation (A12) we

have

zE− = rξ∑

lm

Dlmirl−1(l +m)(l +m− 1)Pm−1

l−1 (ξ)ei(m−1)φ , (B10)

r−Ez = −r√

1− ξ2e−iφ∑

lm

Dlmirl−1(l +m)Pm

l−1(ξ)eimφ . (B11)

Eqs. (B10) , (B11) and the identity ξ(l + m − 1)Pm−1l−1 (ξ) +

√1− ξ2Pm+1

l−1 (ξ) = (l − m +

1)Pm−1l (ξ) leads to

W− =∑

lm

Dlmirl(l +m)

[

ξ(l +m− 1)Pm−1l−1 (ξ) +

√

1− ξ2Pml−1(ξ)

]

ei(m−1)φ ,

=∑

lm

Vlmi

√

√

√

√

(2l + 1)

4π

(l −m)!

(l +m)!rl(l +m)(l −m+ 1)Pm−1

l (ξ)ei(m−1)φ ,

=∑

lm

Vlmirl√

(l +m)(l −m+ 1)Yl,m−1(θ, φ) , (B12)

Using Eqs. (B9), (B12) and the definition qlmi =∫

ρi (~r) rlY ∗

lm (θ, φ) d3~r we get

〈τix〉 =1

4Re

∫

ρ∗i (~r)i(W+ −W−)d3~r ,

= −1

4Re

∑

lm

iVlmi

[

√

(l −m)(l +m+ 1)q∗l,m+1,i +√

(l +m)(l −m+ 1)q∗l,m−1,i

]

.(B13)

From the relation qlmi = −2l+14π

αlmiVlmi between the multipole moment lm induced in particle

i and the corresponding expansion coefficient Vlmi we obtain

〈τix〉 = Re∑

l

iπ

(2l + 1)αli[Sli + S∗

li] , (B14)

where

Sli =l−1∑

m=−l

√

(l −m)(l +m+ 1)qlmiq∗l,m+1,i . (B15)

17

A similar development for the y component of the torque gives

〈τiy〉 = Re∑

l

π

(2l + 1)αli[Sli − S∗

li] . (B16)

The z component of the torque may be rewritten as

〈τiz〉 =1

4iRe

∫

ρ∗i (~r) [r−E+ − r+E−] d3~r . (B17)

Using definitions of r+ and r− and relations (A27) and (A29) for E+ and E− we get

r−E+ − r+E− = −∑

lm

Dlmirl[

√

1− ξ2Pm+1l−1 (ξ) + (l +m)(l +m− 1)

√

1− ξ2Pm−1l−1 (ξ)

]

eimφ.

(B18)

Using (l+m−1)√1− ξ2Pm−1

l−1 (ξ) = ξPml−1(ξ)−Pm

l (ξ), the expression between square brackets

in Eq. (B18) , which we denote by C becomes

C =√

1− ξ2Pm+1l−1 (ξ) + (l +m)

[

ξPml−1(ξ)− Pm

l (ξ)]

,

= (l +m)ξPml−1(ξ) +

√

1− ξ2Pm+1l−1 (ξ)− (l +m)Pm

l (ξ) . (B19)

Since (l +m)ξPml−1(ξ) +

√1− ξ2Pm+1

l−1 (ξ) = (l −m)Pml (ξ) we get

C = (l −m)Pml (ξ)− (l +m)Pm

l (ξ) ,

= −2mPml (ξ) . (B20)

Using Eqs. (B17), (B18) and (B20) we find

〈τiz〉 =1

2Re

∫

ρ∗i (~r)1

2i(r−E+ − r+E−)d

3~r ,

=1

2Re

∫

ρ∗i (~r)1

2i

∑

lm

Vlmirl2mYlm(θ, φ)d

3~r . (B21)

With the definition qlmi =∫

ρi (~r) rlY ∗

lm (θ, φ) d3~r and the relation qlmi = −2l+14π

αlmiVlmi for

eliminating Vlmi we obtain our final result for the z component of the torque

〈τiz〉 =1

2Re

1

i

∑

lm

Vlmimq∗lmi ,

= Re∑

lm

2πi

2l + 1

m

αli

qlmiq∗lmi . (B22)

18

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21

Figures

1.0 1.5 2.0 2.5

-40

-20

0

20

40

60

ℏω [eV]

Fz

[arb

. u

nit

s]

80

FIG. 1: Electric force between two identical gold nanospheres as a function of frequency of the

applied field, with separation 2.005a between their centers. The solid (dashed) curve corresponds to

parallel (perpendicular) excitation calculated including multipoles up to L = 40. The dash-dotted

curve corresponds to the average force calculated for parallel excitation and separation 3a, with an

amplification factor 1000.

22

1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

ℏω [eV]

τ z [

arb

. u

nit

s]

1.0

1.0

FIG. 2: Time averaged torque acting on a particle for a system of two gold nanospheres subjected

to a rotating electric field, as a function of frequency. Separation between their centers are 2.005a

(solid curve) and 3a (dashed curve). Results were obtained including multipoles up to L = 40 and

L = 10, respectively.

23

ℏω [eV]

Fz

[arb

. u

nit

s]

2.42.22.01.8 2.61.6 2.8

20

40

60

0

FIG. 3: Force between two identical gold nanospheres in the parallel configuration as a function of

frequency, with separation 3a between their centers. Solid and dashed curves correspond to the force

calculated from Coulomb’s law and using the derivative of an interaction potential, respectively.

24